1. Promoting the development of procedural flexibility Dr. Jon R. Star Harvard Graduate School of Education jon_star@harvard.edu

2. Acknowledgements Funding provided by the National Science Foundation and the US Department of Education (Institute for Education Sciences) My collaborators Bethany Rittle-Johnson (Vanderbilt) and Kristie Newton (Temple) Fri Jan 14, 2011PDE Conference2

3. Overview of talk Who am I? Let’s talk about algebra Let’s talk about procedural flexibility Let’s talk about comparison Three relatively simple ways that teachers can take advantage of comparison to improve students’ procedural flexibility What can this look like in practice? Fri Jan 14, 2011PDE Conference3

4. Fri Jan 14, 2011PDE Conference4 Who am I? Taught middle and high school mathematics for 6 years –Favorite course to teach was Algebra I –Intrigued and challenged by students’ difficulties in learning algebra PhD in Education and Psychology from University of Michigan –Focused on students’ learning of algebra

5. Fri Jan 14, 2011PDE Conference5 Who am I? Assistant Professor at Graduate School of Education at Harvard for the past 4 years –Teach methods courses for secondary preservice math teachers –Teach graduate courses in students’ learning of school subjects such as math Served on the faculty at Michigan State University for 5 years

6. Fri Jan 14, 2011PDE Conference6 What am I interested in? Students’ strategies for solving problems in mathematics, but especially in algebra What strategies do students use to solve problems? How do students know what strategies to use on unfamiliar problems?

7. Fri Jan 14, 2011PDE Conference7 Why focus on strategies? Addresses many noted difficulties for students in algebra and in mathematics more generally Only apply procedures by rote Strategies easily forgotten or applied incorrectly Don’t understand the strategies that they are using Difficulty in approaching unfamiliar problems

8. Fri Jan 14, 2011PDE Conference8 Knowledge of strategies Superficial knowledge: Knowledge associated with rote learning, inflexibility, reproduction, and trial/error Deep knowledge (procedural flexibility): Knowledge associated with comprehension, abstraction, flexibility, and critical judgment does not understand “what to do and why” does understand “what to do and why”

9. Fri Jan 14, 2011PDE Conference9 Procedural flexibility Strategic, rather than rote, use of procedures Knowledge of multiple strategies for solving a type of problem Ability to choose the most appropriate strategy for a given problem or situation

10. Fri Jan 14, 2011PDE Conference10 Let’s talk about algebra Procedural flexibility has particular relevance for algebra What is algebra? What is it hard for students? Why is it important for students?

11. What is algebra, to me? Algebra is mostly about: –functions? –expressions and equations and gaining fluency with symbols? –exploring and representing relationships between varying quantities? –algebraic reasoning Certainly all of these are important Fri Jan 14, 2011PDE Conference11

12. Why is algebra hard? 1.Lack of clarity among teachers and researchers about what algebra is Hard to design curriculum and effectively teach students if we can’t agree on what algebra is ( More on this in a moment... ) Fri Jan 14, 2011PDE Conference12

13. Why is algebra hard? 2.For parents, consensus about what algebra is: Algebra is about doing mysterious things with the last three letters of the alphabet?! –Most adults don’t use algebra in their daily lives?! –Why should students learn algebra at all?! –Why don’t we focus on other math topics that are more important and relevant to real life?! Difficult for parents to be supportive of our efforts in algebra with this view of the topic Fri Jan 14, 2011PDE Conference13

14. Why is algebra hard? 3.Algebra represents a leap in abstraction for students, and this is hard! Moving from the tools and ways of reasoning used in arithmetic, which often involve counting or working with concrete objects, to the more abstract reasoning and generalization processes involved in algebra, is a long and hard process for many students Fri Jan 14, 2011PDE Conference14

15. Why is algebra so important? 1.Completing an algebra course is linked to success in later math courses, college graduation rates, ability to get certain kinds of jobs, etc. Algebra is considered a critical gatekeeper course, in that taking and passing algebra opens up educational and career doors that would otherwise be closed Fri Jan 14, 2011PDE Conference15

16. Why is algebra so important? 2.Algebra, and the reasoning that is a part of algebra, is a core component of the discipline of mathematics One reason that students take math in school is to get exposure to what mathematics is all about, and algebra and algebraic reasoning play an extremely important role in mathematics Fri Jan 14, 2011PDE Conference16

17. Why is algebra so important? 3.Some also make the argument that students should take algebra because it is useful in daily life We are constantly using algebra in our personal and professional lives, even when we don't know we are using algebra Fri Jan 14, 2011PDE Conference17

18. What is algebra? At present, two main ways of thinking about algebra in the US mathematics curriculum A function-based view An equation-based view Fri Jan 14, 2011PDE Conference18

19. Function-based view of algebra Algebra is primarily about relationships between quantities that vary, how to represent these relationships, and how to use representations to analyze, generalize, and make predictions about these relationships Fri Jan 14, 2011PDE Conference19

20. Function-based example Sarah decided to spend her savings to buy music for her iPod. She has $100 now and spends $10 each week. How much money does she have in her savings account? Fri Jan 14, 2011PDE Conference20 Week$ 0100 190 280 370

21. Equation-based view of algebra Algebra is primarily about modeling relationships with symbols and then using symbols to determine features of these relationships Fri Jan 14, 2011PDE Conference21

22. Equation-based examples Fri Jan 14, 2011PDE Conference22

23. Connecting both views of algebra Algebra is about both of these views, but we've found it difficult to connect function- and equation-based views in our instruction –Also, these two views are separated temporally - function-based views are present in many middle school curricula but equation-based views are present in most high school curricula Even when both views exist simultaneously in an algebra course, they are not well-integrated Fri Jan 14, 2011PDE Conference23

24. Algebra table of contents Fri Jan 14, 2011PDE Conference24

25. Another way to think about algebra Instead of “What is algebra?”, instead ask “What does it mean to understand algebra?” Fri Jan 14, 2011PDE Conference25

26. Understanding in algebra Representations: Symbols, graphs, tables –Representations are useful to analyzing relationships between quantities Understanding in algebra is: –Ability to move fluently between multiple representations –Ability to operate fluently within each representation Fri Jan 14, 2011PDE Conference26

27. Representations Fri Jan 14, 2011PDE Conference27 GraphicalTabularSymbolic y = 5x - 3 xy 0-3 12 27 312

28. Between representations Analyze situations using graphs, tables, and symbols Make connections between graphs, tables, and symbols Fri Jan 14, 201128PDE Conference

29. Example of “between”: slope Fri Jan 14, 2011PDE Conference29 GraphicalTabularSymbolic y = 5x - 3 xy 0-3 12 27 312

30. Example of “between”: y intercept Fri Jan 14, 2011PDE Conference30 GraphicalTabularSymbolic y = 5x - 3 xy 0-3 12 27 312

31. Within representation Know concepts and skills for working with a single representation But also know multiple strategies in a representation, including the ability to select the most appropriate strategies in that representation for a given problem Flexibility within a given representation Fri Jan 14, 201131PDE Conference

32. ‘Within’ example using tables Fri Jan 14, 2011PDE Conference32

33. Fri Jan 14, 2011PDE Conference33 ‘Within’ example #1 using symbols Time to do some math! Solve this equation for x: 3(x + 1) = 15 Can you solve this same equation again, but in a different (better?) way? Now solve this equation for x: 3(x + 1) = 14

34. Fri Jan 14, 2011PDE Conference34 Problem 1 Strategy 1 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 Problem 1 Strategy 2 3(x + 1) = 15 x + 1 = 5 x = 4 Problem 2 Strategy 1 3(x + 1) = 14 3x + 3 = 14 3x = 11 x = 11/3 Problem 2 Strategy 2 3(x + 1) = 14 x + 1 = 14/3 x = 11/3

35. Fri Jan 14, 2011PDE Conference35 So what? We want students to know multiple strategies in a representation, including the ability to select the most appropriate strategies in that representation for a given problem Choosing most appropriate strategy –reduces error –allows for a quicker solution Flexibility is linked to conceptual knowledge –ideas about equivalence and variables Flexibility is linked to transfer performance

36. Fri Jan 14, 2011PDE Conference36 ‘Within’ example #2 using symbols Solve the following problems for x: Problem 1: Problem 2:

37. Fri Jan 14, 2011PDE Conference37 ‘Within’ example #2 Strategy #1: Cross multiplication strategy Strategy #2: Equivalent fractions strategy 7x = ??? 7 times 2 is 14, so 16 times 2 is 32

38. Fri Jan 14, 2011PDE Conference38 ‘Within’ example #2 Strategy #1: Cross multiplication strategy Strategy #2: Unit rate strategy 8x = ??? 8 times 2 is 16, so 14 times 2 is 28

39. Fri Jan 14, 2011PDE Conference39 All of these ‘within’ examples... Involve knowing multiple strategies to approach a class of problems and selecting the most appropriate strategy for a particular problem Move students beyond rote application of a single procedure to more flexible use of multiple strategies Procedural flexibility

40. Fri Jan 14, 2011PDE Conference40 Developing flexibility How can teachers help students to develop procedural flexibility? Ideally, interventions would be –easy to implement –have big impact on students What kinds of small changes in teachers’ practices appear to have a significant impact on students’ flexibility in algebra? I will discuss three - all draw on the benefits of comparison

41. Fri Jan 14, 2011PDE Conference41 Comparison Is a fundamental learning mechanism Strong empirical support for benefits of comparison from cognitive science Helps focus our attention on critical features of what we are trying to learn For example...

42. Buying a camera at Best Buy How do I decide which camera to buy, and how does comparison help? Let’s say I start with price and pick two cameras at the same price that both look OK Fri Jan 14, 2011PDE Conference42 Canon SD-30, $299.99 Canon SD-1000, $299.99

43. Fri Jan 14, 2011PDE Conference43 Look at camera #1 (SD-30) SD-30 Information Canon SD-30, $299.99

44. Fri Jan 14, 2011PDE Conference44 Look at camera #2 (SD-1000) SD-1000 Information Canon SD-1000, $299.99

45. Fri Jan 14, 2011PDE Conference45 Limits of sequential viewing Hard to remember information about the SD-30 when looking at the SD-1000 and vice versa Hard to know which features the cameras are similar on and which they are different on As a result, I am just as likely to be influenced by superficial features (shape and color) than by substantive features

46. Fri Jan 14, 2011PDE Conference46 What if I compare? Comparing these two cameras Canon SD-30, $299.99Canon SD-1000, $299.99

47. Fri Jan 14, 2011PDE Conference47 Comparison Makes it easier for me to see the features where these cameras are the same and different In the case of mathematics learning...

48. Fri Jan 14, 2011PDE Conference48 Comparison helps students... See similarities and differences between problems, representations, and strategies See how and why a strategy works for a particular problem See how and why a representation is particularly useful for answering certain kinds of questions See why some strategies or representations are better than others for certain kinds of problems

49. Fri Jan 14, 2011PDE Conference49 3 little things that really help Three small, interrelated interventions to help students become more successful in algebra: 1.Alter instructional presentation of problems and strategies so that comparison can occur 2.Engage students in conversations that are focused on comparison 3.Provide opportunities for students to generate multiple approaches so that comparison can occur

50. Fri Jan 14, 2011PDE Conference50 1. Instructional presentation Instructional materials and presentation should provide opportunities for students to see multiple problems and strategies at the same time, rather than sequentially Example 1Example 2 p.1 Example 1 Example 2 p.1 p.2

51. Fri Jan 14, 2011PDE Conference51 Can take several formats Display the same problem solved two different ways, side by side (compare strategies) Display two different problems solved in the same way, side by side (compare problem types) Both of these ways of presenting problems are better for student learning than sequential presentation

52. Fri Jan 14, 2011PDE Conference52 Compare strategies example Students see one worked example 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4

53. Fri Jan 14, 2011PDE Conference53 Compare strategies example Followed by another one on a different page (but same problem) Hard to compare these two strategies when presented in this way –Think back to the camera example 3(x + 1) = 15 x + 1 = 5 x = 4

54. Fri Jan 14, 2011PDE Conference54 Compare strategies example But when presented together, we can compare much more easily Can see that these problems are same Can make sense of the innovative use of the divide step –More likely to adopt this strategy 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 3(x + 1) = 15 x + 1 = 5 x = 4

55. Fri Jan 14, 2011PDE Conference55 Important caution The compared examples need to differ on some important dimensions but not be completely different How the examples differ needs to be important and relevant to the problem solving process

56. Fri Jan 14, 2011PDE Conference56 2. Comparison conversations No evidence that altering instructional presentation by itself is sufficient Needs to be combined with providing opportunities for students to compare and evaluate different strategies that they are learning Two types of conversations –similarities and differences in strategies –evaluation of strategies

57. Fri Jan 14, 2011PDE Conference57 Similarities and differences How are these strategies similar? How are they different? How are these strategies related to other strategies that we have used before?

58. Fri Jan 14, 2011PDE Conference58 Evaluation of strategies Which strategy is better for this problem and why? Why is this strategy the most effective (or most efficient, or most elegant, or the best) to use for this problem? On what kinds of problems is this strategy most effective (or most elegant, or the best)?

59. Fri Jan 14, 2011PDE Conference59 Important cautions Avoid “serial sharing” conversations –This does not provide opportunities for comparison Teachers need to provide visual and gestural cues to aid comparison –Evidence from TIMSS that teachers in high performing counties do more of this than US teachers

60. Fri Jan 14, 2011PDE Conference60 3. Multiple approaches Instruction should provide opportunities for students to generate multiple ways to solve the same problem

61. Fri Jan 14, 2011PDE Conference61 Several different formats Students are given one solution to a problem and are asked to generate another, different solution Students are given a problem and asked to solve it in two different ways Whole class discussion where multiple different solutions are generated –“Can anyone think of another, different way to approach this problem that we haven’t seen yet?”

62. Fri Jan 14, 2011PDE Conference62 Link to comparison The act of producing solution methods that are different requires some comparison Presence of multiple solution strategies to the same problem links to other two recommendations –Instructional presentation which shows two or more strategies side by side –Evaluative conversations about the multiple solution strategies

63. Fri Jan 14, 2011PDE Conference63 Important caution Cannot be busy-work! Instructional emphasis is not on the mere generation of multiple methods but on the subsequent conversations and reflections that this task can lead to Differences in generated solutions methods may be trivial or not relevant –Another reason why this task needs to be combined with some sort of evaluative discussion

64. Fri Jan 14, 2011PDE Conference64 Bottom line Three related but relatively small instructional interventions that have a significant impact on student learning –Show multiple examples side by side –Discuss and evaluate multiple strategies –Have students generate multiple ways

65. In sum... It pays to compare! Comparison is linked to gains in procedural flexibility in algebra Three related but relatively small instructional interventions that have a significant impact on student learning –Show multiple examples side by side –Discuss and evaluate multiple strategies –Have students generate multiple ways Fri Jan 14, 2011PDE Conference65

66. What might this look like? We have developed supplemental curriculum materials (“worked example pairs”) that draw upon these interventions and are testing them in algebra classes We are in the midst of a large randomized controlled trial in Massachusetts evaluating the impact of these materials on students’ learning of algebra Fri Jan 14, 2011PDE Conference66

67. Fri Jan 14, 2011PDE Conference67

68. Let’s go to the video! Context –A teacher using very early version of our materials –Private school, small classes, 9th grade Algebra I As you watch video, look for the ways that the teacher makes use of the three instructional recommendations –Show multiple examples side by side –Discuss and evaluate multiple strategies –Have students generate multiple ways Fri Jan 14, 2011PDE Conference68

69. Current study One week professional development institute Approximately 150 worked example pairs that can be used for all topics in the Algebra I curriculum About 40 teachers using these materials, videotaping themselves, giving students tests –An equal number serving as the “comparison group” Preliminary results in a year or so I can’t share curriculum materials yet! Fri Jan 14, 2011PDE Conference69

70. Questions? Comments? Thanks! Dr. Jon R. Star Harvard Graduate School of Education jon_star@harvard.edu For more information: http://gseacademic.harvard.edu/contrastingcases http://isites.harvard.edu/jon_star